show that if the diagonal of a quadrilateral are equal and bisects each other at right angle then it is a square.
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Given :-
- Let ABCD is a quadrilateral in which diagonals AC and BD intersects at O.
- AC =BD,
- OA = OC,
- OB = OD,
- ∠AOB = ∠BOC = ∠COD = ∠DOA = 90°.
To Prove :-
- ABCD is a square.
Proof :-
Consider,
Similarly,
Similarly,
From equation (1), (2) and (3), we concluded that
Since, opposite pair of sides are equal.
Now,
Consider,
Now,
ABCD is a parallelogram,
Since,
- ABCD is a parallelogram and ∠BAD = 90°.
Therefore,
- ABCD is a square.
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